課程名稱︰ 應用代數 課程性質︰ 選修 課程教師︰ 陳文進 開課學院： 電資院 開課系所︰ 資工系 考試日期（年月日）︰2017/05/09 考試時限（分鐘）： 150 試題 : (用 Latex 格式) 1. Find all the positive integers m that make the following congruence true simutaneously: 64 ≡ 13 mod m 2^8 ≡ 1 mod m 2. Find all integral solutions of the equation: 858x + 364y = 78 3. Find all integral solutions of the congruent equation: 56x ≡ 88 mod 96 4. Find the smallest positive integer that satisfies the following congru- ences simutaneously: x ≡ 5 mod 7 x ≡ 7 mod 11 x ≡ 3 mod 13 5. Let S = {a_1a_2a_3a_4 | a_i \in {0,1}, 1≦i≦4}be the set of binary code words of length 4. We define the binary operator ⊕ on S as follows: u = a_1a_2a_3a_4, v = b_1b_2b_3b_4, then u⊕v = c_1c_2c_3c_4 where c_i = a_i + b_i mod 2 (1 ≦ i ≦ 4). (a) Show that (S,⊕) is a group. Give the indetity and the inverr of u = a_1a_2a_3a_4. (b) Show that H = {0000,1010,0101,1111} is a subgroup of S. (c) Find the coset of H in S. 6. G is a group. The center of G is Z = {u \in G |\forall x \in G, ax =xa}. Prove that Z is an normal subgroup of G. 7. G is a group. If o(a) = 2 for all a \in G, a ≠ e, prove that G is a communtative (Abelian) group. 8. G is a commutative group. a,b \in G, o(a) = m, o(b) = n. Find the order of a^nb^m. 9. Let R[x] = {a_0 + a_1 x + ... + a_n x^n | a_i \in R. n ≧ 0} be the set of polynomials with real coefficients, C = {a + bi| a,b \in R} be the set of complex numbers. (a) Show that the mapping φ: R[x] → C ,φ(p(x)) = p(i) is a sur- jective homomorphism from additive group (R[x], +) to additive group (C,+). (b) What is Ker(φ) ? (c) What is the quotient group R[x]/Ker(φ)? (d) The Fundamental Theorem of Group Isomorphism tells us that R[x]/Ker(φ) \simeq C. Given an isomorphism φ': R[x]/ker(φ) → C. --

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